Fun idea
It is a fun idea. I don't think it would work.
I have an even more fun idea: don't tell the player it won't work - let her think it can and have lots of fun when she fingers the innocent.
Rules
The doppelganger can use its action to
polymorph into a Small or Medium humanoid it has seen, or
back into its true form. Its statistics, other than its size, are the
same in each form.
Interestingly, "polymorph" is not defined anywhere in the rules. It only gets its definition from the local context "... into a Small or Medium humanoid ..." and "... each form ..." and normal English usage "having, assuming, or passing through many or various forms, stages, or the like." The fact that there are spells that use the word does not inform the usage here except in a general way - the doppelganger is not using a polymorph spell; it is using the Shapechanger ability.
Doppelgangers are devious shapeshifters that take
on the appearance of other humanoids, throwing off
pursuit or luring victims to their doom with misdirection
and disguise.
"Appearance" means several things, the most relevant being "the state, condition, manner, or style in which a person or object appears; outward look or aspect" - there is no indication that this is limited to visual appearance only: the doppelganger changes its visual, aural, tactile, and chemical appearance (i.e. what it smells and tastes like).
A doppelganger's adopted form
allows it to blend into almost any group or community,
but its transformation doesn't impart languages,
mannerisms, memory, or personality.
So here is what it doesn't do: by inference, it gets everything else.
They assume attractive
male forms and seduce women, leaving them to raise
their progeny.
You don't get much closer to someone than this - presumably they smell good enough to mate with.
Rationale
The doppelganger is a species that has evolved to blend in with any small or medium humanoid species to thrive and reproduce. It can do this so effectively that it can engage in sexual activity without the other party suspecting.
Some of these humanoid species have great senses of smell (e.g. orcs, goblins) some of them keep dogs or other domestic animals with great senses of smell.
If a doppelganger doesn't smell like a member of the species its impersonating; it will be found out.
Acquired Smells
GMJoe wrote in a comment:
I agree with this answer as far as it goes, but because a doppleganger's shape-shifting doesn't impart mannerisms, memory, or personality, it wouldn't necessarily include scents that are a byproduct of those things; So, if a character was in the habit of using a particular perfume, or of playing with their collection of dogs, or of walking through the tanner's district on the way home from work, or eating large amounts of curry, the doppleganger may well smell different - though, only to the extent that the original might smell different after an unusual day.
Since comments don't last and this is a really good point I have added it in.
It is hard to track down a pure RAW explanation, but gauging by the wording in the description for spell-like abilities d20srd
A spell-like ability usually has a limit on how often it can be used (emphasis mine)
It is referring to the specific spell like ability, not spell like abilities in general, therefore one could infer each Sp that listed can be used however many times is specified in the creatures stat block section for special abilities.
Best Answer
According to "Ecology of the Troglodyte" in Dragon #235, it's not that they stink per se, but that they use scent to communicate.
The scents they describe are many and varied; in fact the name of the trog in the piece is "Rose-Boiled Cabbage-Mild Skunk" as his name is a mix of those scents. But to the average human, more than one or two scents at a time just smell like "stank."